Test: Gauss's Theorem - Civil Engineering (CE) MCQ

Concept:
Gauss divergence theorem: 

where,   is called surface integral

  is called volume integral

 is called divergence of F

Calculation:

Given:

V = volume of cube = (1)³

= 3 + 5 + 6 = 14

Concept:

Green’s Theorem:

If s is the surface bounded by a closed curve C, then

Calculation:

Given:

f₂ = (y – sin x), f₁ = cos x

∴ 

∴ a = -4, b = -2

∴ |ab| = |8| = 8

By Gauss-Divergence Theorem

Concept:

According to Gauss Divergence theorem:

Calculation:

Given:

x² + y² + z² = a²

∴ closed surface is a sphere of radius r

∴ 

Concept:

Gauss Divergence Theorem:

The surface integral of the normal component of a vector function 
 taken around a closed surface S is equal to the integral of the divergence of 
 taken over the volume V enclosed by the surface S. Mathematically;

Calculation:

Given:

 and x² + y² + z² = a²

∴ 

From the Gauss divergence theorem;

where ∫VdV represents volume of sphere.

The equation for the surface of sphere is x² + y² + z² = a² , ∴  

∴ the radius of sphere is a.

The volume of the sphere = 

⇒ 

⇒ 

The given surface is x² + y² = 4, z = 0, and z = 3.

Separating the 'y' component of it, we get:

Putting this as a limit in the limits, we get:

= 

= 

= 

= 

= 

Concept:

The position vector 

When z = 0

Circle x² + y² = a² ⇒ x = a cos θ

The line integral is given as:

Calculation:

Now,

x = a cos θ, y = a sin θ

Now:

a = 5 (Given)

∴ 

The Gauss divergence theorem uses divergence operator to convert surface to volume integral. It is used to calculate the volume of the function enclosing the region given.

The divergence theorem for a function F is given by ∫∫ F.dS = ∫∫∫ Div (F).dV. Thus it converts surface to volume integral.

We could parameterise surface and find surface integral, but it is wise to use divergence theorem to get faster results. The divergence theorem is given by ∫∫ F.dS = ∫∫∫ Div (F).dV
Div (3x i + 2y j) = 3 + 2 = 5.
Now the volume integral will be ∫∫∫ 5.dV, where dV is the volume of the sphere 4πr³/3 and r = 3units.
Thus we get 180π.

We know that electric field lines emerge from a positive charge and move towards a negative charge. Now, charge Q, on the face in front of charge q_1, tries to nullify the field lines emerging from charge q_1. So, the charge on the farther face of the conductor dominates and, hence, a force appears to the right of charge q.

The electric field remains same for the plastic plate and the copper plate, as both are considered to be infinite plane sheets. So, it does not matter whether the plate is conducting or non-conducting.
The electric field due to both the plates,

E = σ/ε0

Gauss Divergence theorem:

Green's theorem:

Stokes theorem:

Concept:

According to Gauss Divergence theorem:

Calculation:

Given:

that S is a closed surface:

Gauss Divergence theorem:

Green's theorem:

Stokes theorem: